Commutative rings with finitely generated monoids of fractional ideals

Let R be a commutative ring with identity and let P(R) be the monoid of principal fractional ideals of R. We show that P(R) is finitely generated if and only if ( the integral closure of R) is finitely generated and is finite. Moreover, is a finite direct product of finite local rings, SPIRs, Bezout domains D with P(D) finitely generated, and special Bezout rings S (S is a Bezout ring with a unique minimal prime P, SP is an SPIR, and P(S/P) is finitely generated). Also, P(R) is finitely generated if and only if F*(R), the monoid of finitely generated fractional ideals of R, is finitely generated. We show that the monoid F(R) of fractional ideals of R is finitely generated if and only if the monoid of R-submodules of the total quotient ring of R is finitely generated and characterize the rings for which this is the case.